Infinite presentability of groups and condensation

We describe various classes of infinitely presented groups that are condensation points in the space of marked groups. A well-known class of such groups consists of finitely generated groups admitting an infinite minimal presentation. We introduce here a larger class of condensation groups, called infinitely independently presentable groups, and establish criteria which allow one to infer that a group is infinitely independently presentable. In addition, we construct examples of finitely generated groups with no minimal presentation, among them infinitely presented groups with Cantor-Bendixson rank 1, and we prove that every infinitely presented metabelian group is a condensation group.Comment: 32 pages, no figure. 1->2 Major changes (the 13-page first version, authored by Y.C. and L.G., was entitled "On infinitely presented soluble groups") 2->3 some changes including cuts in Section

Identities in upper triangular tropical matrix semigroups and the bicyclic monoid

We establish necessary and sufficient conditions for a semigroup identity to hold in the monoid of n x n upper triangular tropical matrices, in terms of equivalence of certain tropical polynomials. This leads to an algorithm for checking whether such an identity holds, in time polynomial in the length of the identity and size of the alphabet. It also allows us to answer a question of Izhakian and Margolis, by showing that the identities which hold in the monoid of upper triangular tropical matrices are exactly the same as those which hold in the bicyclic monoid. Our results extend to a broader class of “chain structured tropical matrix semigroups”; we exhibit a faithful representation of the free monogenic inverse semigroup within such a semigroup, which leads also to a representation by upper triangular tropical matrices

Association of BMI, lipid-lowering medication, and age with prevalence of type 2 diabetes in adults with heterozygous familial hypercholesterolaemia: a worldwide cross-sectional study

Background: Statins are the cornerstone treatment for patients with heterozygous familial hypercholesterolaemia but research suggests it could increase the risk of type 2 diabetes in the general population. A low prevalence of type 2 diabetes was reported in some familial hypercholesterolaemia cohorts, raising the question of whether these patients are protected against type 2 diabetes. Obesity is a well known risk factor for the development of type 2 diabetes. We aimed to investigate the associations of known key determinants of type 2 diabetes with its prevalence in people with heterozygous familial hypercholesterolaemia. Methods: This worldwide cross-sectional study used individual-level data from the EAS FHSC registry and included adults older than 18 years with a clinical or genetic diagnosis of heterozygous familial hypercholesterolaemia who had data available on age, BMI, and diabetes status. Those with known or suspected homozygous familial hypercholesterolaemia and type 1 diabetes were excluded. The main outcome was prevalence of type 2 diabetes overall and by WHO region, and in relation to obesity (BMI ≥30·0 kg/m2) and lipid-lowering medication as predictors. The study population was divided into 12 risk categories based on age (tertiles), obesity, and receiving statins, and the risk of type 2 diabetes was investigated using logistic regression. Findings: Among 46 683 adults with individual-level data in the FHSC registry, 24 784 with heterozygous familial hypercholesterolaemia were included in the analysis from 44 countries. 19 818 (80%) had a genetically confirmed diagnosis of heterozygous familial hypercholesterolaemia. Type 2 diabetes prevalence in the total population was 5·7% (1415 of 24 784), with 4·1% (817 of 19 818) in the genetically diagnosed cohort. Higher prevalence of type 2 diabetes was observed in the Eastern Mediterranean (58 [29·9%] of 194), South-East Asia and Western Pacific (214 [12·0%] of 1785), and the Americas (166 [8·5%] of 1955) than in Europe (excluding the Netherlands; 527 [8·0%] of 6579). Advancing age, a higher BMI category (obesity and overweight), and use of lipid-lowering medication were associated with a higher risk of type 2 diabetes, independent of sex and LDL cholesterol. Among the 12 risk categories, the probability of developing type 2 diabetes was higher in people in the highest risk category (aged 55–98 years, with obesity, and receiving statins; OR 74·42 [95% CI 47·04–117·73]) than in those in the lowest risk category (aged 18–38 years, without obesity, and not receiving statins). Those who did not have obesity, even if they were in the upper age tertile and receiving statins, had lower risk of type 2 diabetes (OR 24·42 [15·57–38·31]). The corresponding results in the genetically diagnosed cohort were OR 65·04 (40·67–104·02) for those with obesity in the highest risk category and OR 20·07 (12·73–31·65) for those without obesity. Interpretation: Adults with heterozygous familial hypercholesterolaemia in most WHO regions have a higher type 2 diabetes prevalence than in Europe. Obesity markedly increases the risk of diabetes associated with age and use of statins in these patients. Our results suggest that heterozygous familial hypercholesterolaemia does not protect against type 2 diabetes, hence managing obesity is essential to reduce type 2 diabetes in this patient population. Funding: Pfizer, Amgen, MSD, Sanofi-Aventis, Daiichi-Sankyo, and Regeneron

Cancellative and Malcev presentations for finite Rees index subsemigroups and extensions

It is known that, for semigroups, the property of admitting a finite presentation is preserved on passing to subsemigroups and extensions of finite Rees index. The present paper shows that the same holds true for Malcev, cancellative, left-cancellative and right-cancellative presentations. (A Malcev (respectively, cancellative, left-cancellative, right-cancellative) presentation is a presentation of a special type that can be used to define any group-embeddable (respectively, cancellative, left-cancellative, right-cancellative) semigroup.).Publisher PDFPeer reviewe

About biography of Kunken Jamyang Shadpa

The article is devoted to the description of some facts of biography of Kunken Jamyang ShadpaI Ngawang Tsondu, the author of manuals of Drepung Gomang monastery, the founder of Labrang monastery